Sharper bounds and structural results for minimally nonlinear 0-1 matrices

The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every $P'$ that is contained in $P$ but not equal to $P$. Bounds on the maximum number of ones and the maximum number of columns in a minimally non-linear 0-1 matrix with $k$ rows were found in (CrowdMath, 2018). In this paper, we improve the bound on the maximum number of ones in a minimally non-linear 0-1 matrix with $k$ rows from $5k-3$ to $4k-4$. As a corollary, this improves the upper bound on the number of columns in a minimally non-linear 0-1 matrix with $k$ rows from $4k-2$ to $4k-4$. We also prove that there are not more than four ones in the top and bottom rows of a minimally non-linear matrix and that there are not more than six ones in any other row of a minimally non-linear matrix. Furthermore, we prove that if a minimally non-linear 0-1 matrix has ones in the same row with exactly $d$ columns between them, then within these columns there are at most $2d-1$ rows above and $2d-1$ rows below with ones.


Introduction
The 0-1 matrix M contains 0-1 matrix P if some submatrix of M either equals P or can be turned into P by changing some ones to zeroes. Otherwise we say that M avoids P. The function ex(n, P) is defined as the maximum number of ones in any 0-1 matrix with n rows and n columns that avoids P. This function has been applied to many problems in combinatorics and discrete geometry, including the Stanley-Wilf conjecture [12], the maximum number of unit distances in a convex n-gon [5], and robot navigation problems [13].
The 0-1 matrix extremal function has a linear lower bound of n for all 0-1 matrices except those with all zeroes or just one entry. Füredi and Hajnal's problem, which is only partially answered, is to find all 0-1 matrices P such that ex(n, P) = O(n) [6]. Marcus and Tardos proved that ex(n, P) = O(n) for every permutation matrix P [12], solving the Stanley-Wilf conjecture, and this linear upper bound was later extended to double permutation matrices [7].
One way to approach Füredi and Hajnal's problem is to find all 0-1 matrices on the border of linearity. A 0-1 matrix P is called minimally non-linear if ex(n, P) = ω(n) but ex(n, P ′ ) = O(n) for every P ′ that is contained in P but not equal to P. Keszegh [9] constructed a class H k of 0-1 matrices for which ex(n, H k ) = Θ(n log n) and conjectured the existence of infinitely many minimally non-linear 0-1 matrices contained in the class. This conjecture was confirmed in [7], without actually constructing an infinite family of minimally non-linear 0-1 matrices.
CrowdMath [2] proved that a minimally non-linear 0-1 matrix with k rows has at most 5k − 3 ones and 4k − 2 columns and bounded the number of minimally non-linear 0-1 matrices with k rows, also finding analogous results for ordered bipartite graphs and forbidden sequences. They posed the problems of finding the maximum number of columns in, the maximum number of ones in, and the total number of minimally non-linear 0-1 matrices with k rows.
In this paper, we sharpen all three bounds from the CrowdMath paper. In Section 2, we prove that a minimally non-linear 0-1 matrix with k rows has at most 4k − 4 ones and 4k − 4 columns. As a corollary, this gives an improved bound on the number of minimally non-linear 0-1 matrices with k rows. In Section 3, we prove several structural results about 0-1 matrices, including that there are not more than four ones in the top and bottom rows of a minimally non-linear matrix and that there are not more than six ones in any other row of a minimally nonlinear matrix. We also prove that if a minimally nonlinear 0-1 matrix has ones in the same row with exactly d columns between them, then within these columns there are at most 2d − 1 rows above and 2d − 1 rows below with ones.

Sharper bounds
For many proofs in this paper, we will refer to the patterns R, Q 1 , and S 1 below respectively.
Minimally non-linear 0-1 matrices with k rows have at most 4k − 4 ones.
Proof. We will prove a slightly stronger result. Call a 0-1 matrix P potentially mnl if (1) P avoids S 1 , Q 1 , R, and all of their reflections, (2) P has no column with a single one between two columns with adjacent ones to the left and right, (3) neither the first nor the last column of P has only a single one that is next to a one in an adjacent column, and (4) P has no empty columns. Note that every minimally nonlinear 0-1 matrix is potentially mnl besides S 1 , Q 1 , R, and their reflections. We will prove that every potentially mnl 0-1 matrix with k rows has at most 4k − 4 ones. We prove this by induction. The statement is true for k = 2 by the definition of potentially mnl.
For the inductive step, we introduce some terminology for relationships between rows in a potentially mnl matrix A. We say that row r encompasses row s if r has a one somewhere to the left of the leftmost one of s, and r also has a one somewhere to the right of the rightmost one of s. Note that if r encompasses s, then r has no ones between two ones of s, or else A would contain S 1 or its reflection. Note that if s has a one in a column between two ones of r, then r must encompass s or else A would contain S 1 , Q 1 , or their reflections.
For the inductive hypothesis, suppose that every potentially mnl 0-1 matrix with k rows has at most 4k − 4 ones, and let A be a potentially mnl 0-1 matrix with k + 1 rows. Name the rows r 1 , . . . , r k+1 so that r k+1 is a row that does not encompass any other row, it has the rightmost one among such rows, and it has the fewest ones among such non-encompassing rows with the rightmost one.
Note that r k+1 must have all of its ones in adjacent columns in A, or else r k+1 would encompass another row. By construction, r k+1 has at most two ones. If we remove r k+1 from A to obtain A ′ , then A ′ is not necessarily potentially mnl since there could be empty column(s) or violation of (2) or (3), but not both. So we fix whatever is present. If we remove empty column(s) then violation of (2) or (3) may be introduced, so we fix it too, but empty column doesn't show up after fixing violation of (2) or (3). So the process ends.
Introduced either way, violation of (2) or (3) are always in consecutive and at most two columns. When we remove them, there will be no more violation of (2) or (3) in any other columns. The number of ones deleted is bounded by 2 + 2 = 4.
The fact that a minimally non-linear 0-1 matrix with k rows has at most 4k − 4 columns follows immediately from the last result. Proof. This follows from the last result since a minimally non-linear 0-1 matrix cannot have any empty columns [19,9].
We obtain a slight improvement on the upper bound on the number of minimally non-linear 0-1 matrices with k rows in [2] using the last corollary.

Corollary 2. For k > 2, the number of minimally non-linear 0-1 matrices with k rows is at most
We also mention a few results about extremal functions of ordered graphs that are proved the same way as the results for minimally non-linear 0-1 matrices. These results improve bounds from [2].

Structural results
In this section, we find bounds on the number of ones in each row of a minimally non-linear 0-1 matrix, in addition to other structural results about minimally nonlinear 0-1 matrices. For a few of the results in this section, we will refer to the patterns Q 3 and S 2 respectively below.
We start with a surprising lemma about the structure of minimally non-linear patterns that is based on Q 1 , S 1 , S 2 , their reflections, and rotations. : M is a minimally non-linear 0-1

Lemma 1. Cross lemma
Proof. If not, M contains S 1 , S 2 , Q 1 , Q 3 , or one of their reflections or rotations.
The next two lemmas allow us to obtain bounds on the number of ones in each row of a minimally non-linear 0-1 matrix.  1. There exists a row r ′ above r with some 1 between columns c + 1 and d − 1 inclusive, and every row above r ′ also has some 1 between columns c + 1 and d − 1 inclusive.

Lemma 2. Suppose that M is a minimally non-linear h
2. There exists a row r ′ below r with some 1 between columns c + 1 and d − 1 inclusive, and every row below r ′ also has some 1 between columns c + 1 and d − 1 inclusive.
As a corollary, we can derive bounds on the number of ones in rows of a minimally non-linear 0-1 matrix.

Lemma 4. There cannot be more than four ones in the bottom (or top) row of a minimally non-linear matrix M.
Proof. If there are in the bottom row, let them be in columns x < y < z < u < v. By Lemma 2, M 1,a = M 1,b = 1 where x < a < z and z < b < v, and M contains S 1 , contradiction.
Lemma 5. There cannot be more than six ones in any row of a minimally nonlinear matrix M.
Proof. If there are in some row of M that is not the top or bottom, let them be in columns x < y < z < u < v < w < t. By Lemma 3, M has ones in columns c, d, e, each in the first or last row, with x < c < z, z < d < v, and v < e < t. Thus M contains S 1 or its reflection, a contradiction.
The next few lemmas allow us to describe the structure of the ones in the top and bottom rows of a minimally non-linear 0-1 matrix. Proof. At least two of the three ones in bottom row are adjacent, otherwise we have three mutually non-adjacent ones. If these columns are numbered x, x + Together, the next two results show that if a minimally non-linear 0-1 matrix has ones in the same row with exactly d columns between them, then within these columns there are at most 2d − 1 rows above and 2d − 1 rows below with ones. Proof. Consider the intersection of the d columns and all the rows above. Each column has no more than two ones according to Lemma 9, and no column has non-adjacent ones. If the top 2d rows are all non-empty, then every column has exactly two ones, and every row has exactly a one. Moreover the top two rows have ones in the same column, making M still non-linear when the top row is removed, contradiction.